Optimizing Sharpe Ratio: Risk-Adjusted Decision-Making in Multi-Armed Bandits

TL;DR

This paper introduces novel algorithms for optimizing the Sharpe Ratio in multi-armed bandit settings, addressing risk-adjusted decision-making in finance with theoretical guarantees and superior empirical performance.

Contribution

It proposes the UCB-RSSR algorithm for regret minimization and new BAI algorithms, providing regret bounds and error probabilities, advancing risk-aware portfolio optimization methods.

Findings

UCB-RSSR achieves O(log n) regret for two-armed bandits.

Proposed BAI algorithms outperform existing methods in various setups.

Algorithms demonstrate practical effectiveness in risk-aware portfolio management.

Abstract

Sharpe Ratio (SR) is a critical parameter in characterizing financial time series as it jointly considers the reward and the volatility of any stock/portfolio through its variance. Deriving online algorithms for optimizing the SR is particularly challenging since even offline policies experience constant regret with respect to the best expert Even-Dar et al (2006). Thus, instead of optimizing the usual definition of SR, we optimize regularized square SR (RSSR). We consider two settings for the RSSR, Regret Minimization (RM) and Best Arm Identification (BAI). In this regard, we propose a novel multi-armed bandit (MAB) algorithm for RM called UCB-RSSR for RSSR maximization. We derive a path-dependent concentration bound for the estimate of the RSSR. Based on that, we derive the regret guarantees of UCB-RSSR and show that it evolves as O(log n) for the two-armed bandit case played for a horizon n. We also consider a fixed budget setting for well-known BAI algorithms, i.e., sequential halving and successive rejects, and propose SHVV, SHSR, and SuRSR algorithms. We derive the upper bound for the error probability of all proposed BAI algorithms. We demonstrate that UCB-RSSR outperforms the only other known SR optimizing bandit algorithm, U-UCB Cassel et al (2023). We also establish its efficacy with respect to other benchmarks derived from the GRA-UCB and MVTS algorithms. We further demonstrate the performance of proposed BAI algorithms for multiple different setups. Our research highlights that our proposed algorithms will find extensive applications in risk-aware portfolio management problems. Consequently, our research highlights that our proposed algorithms will find extensive applications in risk-aware portfolio management problems.